A grid cell is a place-modulated neuron whose multiple firing locations define a periodic triangular array covering the entire available surface of an open two-dimensional environment. Grid cells are thought to form an essential part of the brain’s coordinate system for metric navigation. They have attracted attention because the crystal-like structure underlying their firing fields is not imported from the outside world, but created within the nervous system. Understanding the origin and properties of grid cells is an attractive challenge for anybody wanting to know how brain circuits compute.

History

The experimental study of spatial representations in the brain entered a new era with the discovery of place cells in the 1970s (O’Keefe & Dostrovsky, 1971) and head-direction cells in the 1980s (Ranck, 1985; Taube et al., 1990). Discharging whenever the rat is in a certain place in the environment, or whenever the rat’s head points in a certain direction, these cells are likely to contribute to the brain’s systems for local navigation. Soon after their discovery, place cells were suggested to be the elements of a neural representation of allocentric space and the animal’s own position within it which the animal could use to find its way from one place to another (O’Keefe & Nadel, 1978; McNaughton et al., 1996). The representation was thought to form a map-like framework for storage of experience in memory, a framework referred to as the 'cognitive map' (O'Keefe & Nadel, 1978). During the years that followed, the conception of the hippocampus as a single dynamic spatial map was challenged by accumulating evidence suggesting that the very same place cells participate in a number of representations, even in the same spatial location (Bostock et al., 1991). Researchers also showed that place-specific firing persisted in CA1 neurons even after removal of all intrahippocampal input from CA3 (Brun et al., 2002). Because only direct perforant-path inputs from the entorhinal cortex were left by these lesions, it was concluded that spatial signals were either computed either in CA1 itself or, more likely, in a general metric navigational system upstream of the hippocampus (Brun et al., 2002), a possibility raised early on (O’Keefe, 1976).
In response to these challenges, Fyhn and colleagues (2004) recorded directly from layers II and III of the medial entorhinal cortex, in the area that provided the strongest projections to the place cells of the dorsal hippocampus (Witter et al., 1989; Dolorfo & Amaral, 1998). Principal neurons in this area had multiple sharply tuned firing fields which collectively signalled the rat’s changing position with a precision similar to that of place cells in the hippocampus. When these cells were recorded in sufficiently large two-dimensional environments, it became clear that the many fields of each neuron formed a periodic triangular array, or a ‘’’grid’’’, that tiled the entirety of the animal's environment, like the cross-points of graph paper (Hafting et al., 2005) (Figure 1).

Figure 2: Firing fields of cells at different positions along the dorsomedial-to-ventrolateral axis of the medial entorhinal cortex. The figure shows a parvalbumin-stained sagittal brain section with the dorsocaudal parts of the medial entorhinal cortex (heavily stained). Representative grid fields are arranged along the dorso-ventral axis where they were recorded. Grid scale (spacing and field size) increases with distance from the dorsomedial border with the postrhinal cortex (arrow).

Grid cells and neural circuits for path integration

A key property of the entorhinal representation is its apparently universal nature (Hafting et al., 2005; Fyhn et al., 2007; see Redish & Touretzky, 1997 and Sharp, 1999, for theoretical suggestions consistent with this observation). Unlike place cells in the hippocampus, the entorhinal grid map is activated in a stereotypic manner across environments, irrespective of the environment's particular landmarks, suggesting that the same neural map is applied wherever the animal is walking. The rigid structure of the map and its independence of particular landmarks suggest that firing positions must be integrated in these cells from speed and direction signals, without reference to the external environment. This process is referred to as ‘’’path integration’’’. While path integration is likely to determine the basic structure of the dynamic firing matrix, the stability of grid vertices and grid orientations relative to cues in the external environment implies that the grid map must be associated with geometric boundaries and landmarks (Hafting et al., 2005). Under certain conditions, such as when the boundaries of a familiar recording box are displaced slightly, these associations may override the concurrent path integration-driven processes (Barry et al., 2007). How path integration signals are integrated with external sensory input has not been determined.

Figure 3: Firing fields of three co-localized grid cells recorded simultaneously while a rat ran around in a large circular enclosure (2 m diameter). Left: The rat's trajectory is shown in black; spike locations for each cell (t1c1, t2c1 and t2c2) are shown in blue, red, and green, respectively. Middle: Peak firing locations for each of the three cells. Right: Peaks are offset to visualize difference in spatial phase but similarity in spacing and orientation. Reproduced, with permission, from Hafting et al. (2005).

An entorhinal spatial map

The grid of each cell can be described by three parameters:

spacing (the distance between fields),

orientation (the tilt of the grid relative to a reference axis), and

spatial phase (displacement in the x and y directions relative to an external reference point).

These parameters are mapped differently onto the entorhinal map (Hafting et al., 2005). Neighbouring cells in layer II of medial entorhinal cortex have similar spacing and orientation. Their spacing increases monotonically from the dorsomedial pole of the medial entorhinal cortex to more ventrolateral positions of the same area (Figure 2.), just like the spatial scale of place cells increases from the dorsal to the ventral pole of the hippocampus (Jung et al., 1994; Kjelstrup et al., 2007). Whether grid cells are organized with respect to orientation has not been firmly determined, although preliminary observations suggest that grid cells in different regions of the left and right medial entorhinal cortex may have different orientation (Hafting et al., 2005; Fyhn et al., 2007). In contrast to spacing and orientation, the spatial phase of the grid is distributed, i.e. the firing vertices of co-localized grid cells are shifted randomly (Figure 3.), in the same way that neighbouring place cells in the hippocampus have different place fields. The functional significance of this mixed topographic-nontopographic organization has not been established. A key objective for future research will be to determine whether the entorhinal spatial map has discrete subdivisions. Architectonic features of the entorhinal cortex are suggestive of a modular organization (Witter & Moser, 2006), but whether the apparent anatomical cell clusters of this brain region correspond to functionally segregated grid maps, each with their own spacing and orientation, remains to be determined.

Figure 4: Model for transformation of periodic grid fields to non-periodic place fields. Place cells receive input from grid cells with similar spatial phase (a common central peak) but a diversity of spacings and orientations. Connection weights are indicated by the thickness of the arrows. Interneurons (red) provide non-specific inhibition. Colour code as in Figure 1. Adapted, with permission, from Solstad et al. (2006).

The relation between grid cells and place cells

The majority of the principal cells in layers II and III of medial entorhinal cortex have grid properties (Sargolini et al., 2006). Thus, most of the spatially selective cortical input to hippocampal place cells is likely to originate from entorhinal grid cells. An important question raised by this possibility is how the periodic spatial firing pattern of the grid cells is transformed to a non-periodic signal in place cells. If all grid cells projecting to a particular place cell had a single, common scale, the hippocampal place field would be expected to repeat itself at intervals similar to the grid spacing. If the inputs vary in spacing and orientation, however, linear summation would result in firing fields with very large repetition cycles, much beyond the scale of any laboratory environment (O’Keefe & Burgess, 2005; Fuhs & Touretzky, 2006; McNaughton et al., 2006; Solstad et al., 2006) (Figure 4.). In such cells, only a single field would be seen in standard recording boxes. Because a given location in the hippocampus may receive inputs from more than 25% of the dorsomedial-to-ventrolateral axis of the medial entorhinal cortex (Witter et al., 1989; Dolorfo & Amaral, 1998), the latter possibility is more likely.

One potential concern for these models is the observation that the set of grids in a given hemisphere may cover only a single orientation and a discrete set of scales (Barry et al., 2007). This might impede their simple summation to form place cells, and suggests that grids might implement something analogous to modulo arithmetic to represent location over a large scale with high efficiency (Burak et al., 2006; Gorchetchnikov & Grossberg, 2007). Whether the grid cell population has multiple orientations and whether scales are continuous or discontinuous needs further experimental study, however.

The above models also rely on the unproven assumption that the spatial phase of the contributing grid cells overlaps significantly. This assumption may not be necessary. Place fields could be generated merely from random connectivity by a competitive Hebbian learning process, provided that there is enough variability in orientation, phase and spacing of the afferent grid cell population (Rolls et al., 2006). Current experimental data are not conclusive as to whether and to what extent plasticity is required for grid to place transformations. While synaptic plasticity is necessary for shaping and stabilizing the hippocampal spatial representation in a novel environment, modifications in the connectivity matrix may not be required for manifestation of place-specific firing as such. Spatially confined firing fields can still be seen in CA1 pyramidal neurons after deletion of NMDA receptors in CA3 or CA1 (McHugh et al., 1996; Kentros et al., 1998; Nakazawa et al., 2002), suggesting that, at short time scales, place fields can be generated and maintained by pre-existing connections or by non-NMDA-receptor-dependent plasticity, at least in some hippocampal areas.

Models of grid cell formation

A common feature of published models of grid field formation is the assumption that neurons in the medial entorhinal cortex participate in path integration-based computations based on speed and direction signals from specialized input cells but calibrated against external sensory inputs in order to correct for cumulative error. It is possible to categorize the models into those that propose the processing to take place at the population level by virtue of the recurrent connectivity of the network and those in which the path integration is performed at the single neuron level.

The network models show how spatial representation can emerge from a ‘’’continuous attractor network’’’, a manifold of stable states which permits smooth variation of a spontaneously generated representation in accordance with the trajectory of the rat (Tsodyks & Sejnowski, 1995). Given a certain level of global inhibition in a network with recurrent connectivity, initially random patterns of activity will spontaneously give rise to an organized ‘bump’ of activity centred on mutually connected cells which, in the case of the grid cells, may have a common set of firing vertices (Fuhs & Touretzky, 2006; McNaughton, 2006). The activity bump may be envisaged to move between grid cells with different vertices as the animal moves from one place to another in a two-dimensional environment. The translation is thought to depend on a path integration mechanism where changes in direction and speed continuously modulate the effective connectivity between cells (Zhang, 1996; Skaggs et al., 1995). In the first model that was published after the discovery of grid cells, Fuhs and Touretzky (2006) modelled the entorhinal cortex as a topographically organized network where neighbouring cells had similar grid phases, such that each place in the environment was represented as a grid pattern on the cell layer. When the animal moves, this ‘grid skeleton’ was thought to be rigidly translated across the entorhinal cell surface, based on inputs that were proportional to the animal’s speed in a preferred direction, which was different for different neurons. Facing the observation that neighbouring grid cells apparently do not have similar grid phases (Hafting et al., 2005), a non-topographical model was developed by McNaughton and colleagues (2006). In this model, a topographical network serves only as a tutor to train, during development, several smaller clusters of grid cells with randomly distributed Hebbian connections and no topographical organization. The inputs from the tutor are scrambled, such that neurons with similar phase may not necessarily become neighbours, but they will be associated by synaptic plasticity. Because the tutor has the periodicity of a grid, the synaptic matrix of the developing entorhinal network becomes toroidal. The toroidal connectivity is thought to account for the spatial periodicity of the grid pattern during movement. To displace the representations in accordance with the animal’s trajectory, this model introduces an additional layer of cells whose firing is modulated by place, head direction and speed. Neurons with such conjunctive properties have been identified in layers III-VI of the medial entorhinal cortex (Sargolini et al., 2006).

In the single-neuron models, a common starting point is that grid cells, like hippocampal place cells, exhibit ‘’‘theta phase precession’’’, a progressive advance in firing during the theta cycle as the animal moves through an individual firing field (O’Keefe & Recce, 1993; Hafting et al., 2006). Phase precession has been modelled as the result of interference between two oscillatory signals with theta frequencies differing in proportion to the animal’s speed (O’Keefe & Recce, 1993). The interference pattern can be decomposed into a fast oscillation that advances with respect to the theta rhythm, and a slower modulation with a phase that integrates the speed of the rat and thus reflects its position along the trajectory. In two dimensions, the interference could be viewed as the interaction of a somatic intrinsic oscillator at theta frequency with several dendritic oscillators, whose frequencies reflect the projection of the rat velocity in some characteristic preferred direction (Burgess et al., 2007). When the slow modulation of these linear interference patterns are combined, and their directions differ in multiples of 60 degrees, a triangular grid map is obtained. In a conceptually similar model, grid fields have been proposed to emerge as Moiré interference patterns from two theta-grid fields (Blair et al., 2007). The key element of this model, the theta-grid cell, has not yet been found in the rat brain.

Future

Grid cells may be the key elements of a modularly organized network for metric spatial representation. The neural map is thought to represent self-location based on integration of speed and direction signals during movement. While the contours of some possible mechanisms can be envisaged, much work is needed to understand how the grid pattern is generated in single cells and in neural networks of the entorhinal cortex and beyond. With the introduction of new tools for genetic activation and inactivation of particular circuit components, it may be possible to decipher the underlying mechanisms.

Gorchetchnikov, A. and Grossberg, S. (2007) Space, time and learning in the hippocampus: How fine spatial and temporal scales are expanded into population codes for behavioral control. Neural Netw. 20, 182–93. doi:10.1016/j.neunet.2006.11.007.